There's a function on your camera that allows you to adjust the exposure, the amount of light entering the camera lens over a period of time. Using long exposures for taking portraits or macro shots is a completely horrid idea, but the prolonged shutter speed does have some rather lovely perks if you know how and when to use it.
The most common ways to use long exposures is in the dark with a bit of light that you may wave around to create awesome shapes and streaks, in the big city to capture the movement of traffic, or creating a misty waterscape:
The most essential item you'll need to do this besides a camera with an adjustable shutter speed is a tripod. If you don't have a tripod when you take long exposures, then you will have nothing focused upon no matter how stable you think your hands are, and the photo will be a flop. That is, unless you're doing something along the lines of this where it's all about the light shapes and nothing else:
Just be careful when you're creating light streaks with a laser pointer, flashlight, etc, to move around a lot so that you're not caught on the film/photo like the E=MC2 photo above. Although, that's just a matter of taste, obviously, whether or not you'd like to be seen.
As for your exact shutter speed, it really depends on the circumstances. Just experiment and find what works best for your situation. If that's not enough info, I suggest checking out this chart to help you determine the shutter speed and ISO setting for the best producible photograph.
Another photography post: The Tilt-Shift Effect and Bokeh